NSF-BSF: AF: Collaborative Research: Small: Randomized preconditioning of iterative processes: Theory and practice

预测股票投资组合长期走势的一种简单方法是绘制过去几个月跟踪的每日价格图表，然后在图表中拟合一条线。这条线的斜率可以帮助预测价格的趋势是上涨还是下跌。这是所谓“回归问题”的最基本的版本。回归问题以更复杂的形式存在于许多领域，包括金融、统计学、科学、遗传学和工程学；当涉及到对事件的及时诊断和预测时，它们的快速解决方案是必须的。该项目旨在通过加速器(称为预条件化程序)加快回归问题的解决。加速器的设置速度非常快，从最初的问题中随机挑选几块：想着扔骰子来决定下一步要选什么。尽管这听起来可能很随意，但它是有效的，因为回归问题往往有很多冗余和重复，使得很难错过重要的部分。此外，这是加速回归问题的一种安全方法：在随机化产生一些低效加速器的可能性很小的情况下，我们仍然可以：加速器速度较慢，但我们仍在解决原始问题--只是速度没有预期的那么快。该项目涉及在各种实际环境中加速回归问题的设计和分析，特别关注人类基因。线性最小二乘/回归问题在许多计算科学中是最重要的，或者是独立的，或者作为优化方法的外部迭代中的序列的一部分。该项目涉及通过“动态的”随机化预条件来加速回归问题的解决，该预条件可以在求解器的内部迭代中改变，或者在优化方法的外部迭代中改变在最小二乘问题上。具体的优化方法包括：(I)用于求解广义线性模型的迭代重加权最小二乘法；(Ii)用于线性规划的内点方法；(Iii)用于训练过参数神经网络的非线性最小二乘法；以及(Iv)用于计算低阶张量分解的高阶正交迭代。该项目将影响许多领域，并特别关注人类遗传学。这些方法的有效性将在标准测试套件以及来自英国生物库数据库的大规模矩阵上得到验证。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

A simple way to predict the long-term movement of your stock portfolio is to graph the daily prices tracked over the past several months, and then fit a line through the graph. The slope of the line can help to predict whether the trend is for prices to increase or to decrease. This is the most basic version of a so-called `regression problem'. Regression problems, in more sophisticated forms, are mainstays in a multitude of areas, including finance, statistics, science, genetics, and engineering; and their fast solution is a must when it comes to timely diagnoses and prediction of events. The project aims to speed up the solution of regression problems through accelerators (called 'preconditioners'). The accelerators are set up very fast, by picking and choosing a few pieces at 'random' from the original problem: Think of throwing dice to determine what to pick next. Although this may sound haphazard, it is efficient because regression problems tend to have a lot of redundancy and repetition, making it difficult to miss an important piece. In addition, this is a safe way of accelerating regression problems: On the off-chance that the randomization should produce somewhat inefficient accelerators, we are still ok: The accelerator is slower, but we are still solving the original problem --just not quite as fast as expected. The project involves the design and analysis of accelerators for speeding up regression problems in a variety of practical settings, with particular attention to human genetics.Linear least squares/regression problems are of primary importance in many computational sciences, either standalone on their own or as part of a sequence in the outer iterations of an optimization method. The project involves accelerating the solution of regression problems via `dynamic' randomized preconditioners that can either change across inner iterations of a solver or else change across least squares problems in outer iterations of an optimization method. Specific optimization methods to be investigated include: (i) iteratively reweighted least squares for solving generalized linear models; (ii) interior point methods for linear programs; (iii) nonlinear least squares for training overparameterized neural networks; and (iv) high-order order orthogonal iteration for computing low-rank tensor decompositions. The project will impact many domains and pay particular attention to human genetics. The effectiveness of the methods will be validated on standard test suites, as well as large-scale matrices from the UK Biobank dataset.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.